Question

y
sin 20° sin 40° sin 60° sin 80º = =​

Answer 1

Step-by-step explanation:

■ we know that

●2sinAsinB = cos (A-B) - cos (A+B)

●2cosAsinB = sin (A+B) - sin(A-B)

● sin(180-A) = sin A

now ,

■ sin20°•sin40°•sin60°•sin80°

$= \frac{1}{2} [2 sin20sin40] \frac{ \sqrt{3} }{2} \sin(80) \\ = \frac{ \sqrt{3} }{4} [ \cos(20 - 40) - \cos(20 + 40) ] \sin(80) \\ = \frac{ \sqrt{3} }{4} [ \cos( - 20) - \cos(60) ] \sin(80) \\ = \frac{ \sqrt{3} }{4} [ \cos(20) - \frac{1}{2} ] \sin(80) \\ = \frac{ \sqrt{3} }{4} [ \cos(20) \sin(80) - \frac{1}{2} \sin(80) ] \\ = \frac{ \sqrt{3} }{4} [ \frac{1}{2} (2 \cos(20) \sin(80) ) - \frac{1}{2} \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [2 \cos(20) \sin(80) - \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [ \sin(20 + 80) - \sin(20 - 80) - \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [ \sin(100) - \sin( - 60) - \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [ \sin(180 - 80) - \sin( - 60) - \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [ \sin(80) - \sin( - 60) - \sin(80) ] \\ = \frac{ \sqrt{3} }{8} [ - \sin( - 60) ] \\ = \frac{ \sqrt{3} }{8} [ \sin(60) ] \\ = \frac{ \sqrt{3} }{8} [ \frac{ \sqrt{3} }{2} ] \\ = \frac{3}{16}$

● keep smiling &

always shining.............:)